


The seven (thankfully) theoretical bridges of Nargothrond

by Beleriandings



Series: Nargothrond and Beyond [4]
Category: The Silmarillion and other histories of Middle-Earth - J. R. R. Tolkien
Genre: Gen, Mathematics, Sass and Snark, silliness
Language: English
Status: Completed
Published: 2015-05-25
Updated: 2015-05-25
Packaged: 2018-04-01 05:20:59
Rating: General Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 940
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/4007404
Author URL: https://archiveofourown.org/users/Beleriandings/pseuds/Beleriandings
Summary: <blockquote class="userstuff">
              <p>Celebrimbor sets the mysterious newcomer in Nargothrond an interesting problem to think over. After all, he does seem to like bridges.</p>
            </blockquote>





	The seven (thankfully) theoretical bridges of Nargothrond

**Author's Note:**

> A somewhat silly outtake from [Between These Carven Walls](http://archiveofourown.org/works/3718783?view_full_work=true) [chapter 6](http://archiveofourown.org/works/3718783/chapters/8637415), though it stands alone. For background, here's the Wikipedia article on the [Seven Bridges of Königsburg](http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg), on which this is based, if it’ll help you visualise it.

“Do you know the place where the Ringwil joins the Narog?” Celebrimbor asked Agarwaen. “Where the two rivers become one? It makes the shape of a tuning fork, just so.” He drew it upon a piece of paper, with straight lines for the rivers.

“Of course. But it doesn’t look like that.”

“No, but that’s not the point.” Celebrimbor merely sighed at the stare he received in answer to this. “Now, say that in between those branches is an islet…” he drew another line across the narrow end of the branching, separating off a triangle of land. “Now - ”

“There is no such islet” interrupted the man, narrowing his eyes.

“Say that there is.”

“But it doesn’t exist!”

Celebrimbor rolled his eyes. “Then say it’s a different river. A river that is not real. It’s got a west bank, a south bank, an islet, and a northern peninsula, just so.”

“What purpose does this serve?”

“Just listen. If there are two bridges to the islet from west bank, and two from the east bank…”

“There is only going to be one bridge.”

“It’s an  _imaginary_  river, remember?”

Agarwaen pursed his lips.

“There are two bridges from the western shore, and two from the eastern shore, and all lead to the islet. Then there is another bridge joining the islet to the peninsula to the north…” he marked the bridges on his crude drawing as crosses in pencil, “…and one more on each side of the northern peninsula, joining it to the west and east banks.” He looked up at Agarwaen, who was squinting at the map he had drawn. “So; find me a way to visit the northern peninsula, the west and east banks, and the islet, without crossing the same bridge more than once.”

Agarwaen blinked. “That is easy, you would just need to…” he frowned, trying to trace a path with his finger. “No, I mean… wait, that wouldn’t work…”

Celebrimbor smiled. “Keep trying.”

“What purpose does this serve?”

“Oh, a very important bridge-engineering purpose, I promise. You might want to get back to me on it.”

——–

“It’s not possible” said Agarwaen, bursting into Celebrimbor’s study several days later, his voice heated. “There’s no  _way_  it’s possible, you  _always_  have to cross a bridge more than once. Damn you, you’ve been deliberately wasting my time, Curufinwion.”

Celebrimbor didn’t even bother to correct him on the name. “It’s not a waste of time” he said, coolly, “if you know  _why_  it’s not possible.”

Agarwaen wavered, frustration and curiosity evidently warring within him. “I suppose you’re going to explain now then” he said, sullenly.

Celebrimbor nodded, taking the diagram back, which had been much drawn on and clearly crumpled and flattened once more. He picked up his pen and dipped it in red ink, the kind used for marking maps. “You see, you have to highlight the bridges, not the islands” he said, drawing a thin red line across each bridge, placing a dot on each island and connecting them. Then he made the dots larger, ink bleeding through the paper.

Agarwaen frowned. “What purpose does that serve?”

“How many lines - bridges - does each island connect to?”

“Well, those three of them connect to three bridges, and that one to five…”

Celebrimbor nodded. “And what does that tell us?”

“It tells us…” Agarwaen furrowed his brow.

“We know that every time you enter a dot - an island - you must leave by a different bridge than you came in by. And why is that significant?”

Celebrimbor watched as Agarwaen traced the paths with his finger. “You’ll run out of bridges” he said at last. “There are three bridges going from that point, but if two are already used, you’ll be stuck.”

“Exactly. It has to be an even number” said Celebrimbor. “Do you understand, then, why you were having so much trouble?”

“I thought it was just the usual way of things, for me” he muttered.

“No, not this time. This time it was not the - well, whatever it is, that seems to cause everything you touch to go up in flames.”

“Excuse me!”

“No, what I meant to say is that it is  _fundamentally_  a problem, and that the reason you couldn’t solve it was because it _couldn’t_  be solved with that configuration of bridges.”

“Are you trying to tell me we should be building  _more_  bridges in Nargothrond?”

“I meant it as nothing of the sort” said Celebrimbor. He cringed a little. “Valar forbid. But as you said, this configuration is unlike Nargothrond’s rivers, unless we build an islet too, between the Narog and the Ringwil.”

“Do you think that would be a good idea?”

“Not particularly. And besides, I don’t see why one would need such a property in their arrangement of bridges. Not terribly useful defensively…”

“Then why even bring it up? Honestly, you people, sometimes I think I will never understand…”

“Don’t your people delight in such problems merely for the sake of them?”

“I have not known anything like  _delight_  in the course of my labours over this, I must say” said Agarwaen archly.

“Your loss.”

Agarwaen’s only answer to this was a scowl of frustration. He turned to sweep from the room, but as he did so the corner of his cloak caught the ink pot, spilling a spreading red stain all over the crumpled paper on the table.

 _Well_ , thought Celebrimbor,  _they warned me he was clumsy and that accidents seemed to happen around him, but that was some truly worrying symbolism_.

With a sigh, he put it from his mind and went back to work in salvaging his structural drawings for the bridge.


End file.
